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Light-Matter Interactions and Quantum Optics

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Light-Matter Interactions and Quantum Optics Light-Matter Interactions and Quantum Optics. Jonathan Keeling http://www.tcm.phy.cam.ac.uk/~jmjk2/qo Contents Contents iii Introduction v 1 Quantisation of electromagnetism 1 1.1 Revision: Lagrangian for electromagnetism . 2 1.2 Eliminating redundant variables . 3 1.3 Canonical quantisation; photon modes . 5 1.4 Approximations of light-matter coupling . 6 1.5 Further reading . 7 2 Quantum electrodynamics in other gauges 9 2.1 Freedom of choice of gauge and classical equations . 9 2.2 Transformation to the electric dipole gauge . 11 2.3 Electric dipole gauge for semiclassical problems . 15 2.4 Pitfalls of perturbation . 17 2.5 Further reading . 18 3 Jaynes Cummings model 19 3.1 Semiclassical limit . 19 3.2 Single mode quantum model . 20 3.3 Many mode quantum model | irreversible decay . 23 3.A Further properties of collapse and revival . 27 4 Density matrices for 2 level systems 29 4.1 Density matrix equation for relaxation of two-level system . 29 4.2 Dephasing in addition to relaxation . 33 4.3 Power broadening of absorption . 36 4.4 Further reading . 37 5 Resonance Fluorescence 39 5.1 Spectrum of emission into a reservoir . 39 5.2 Quantum regression \theorem" . 40 5.3 Resonance fluorescence spectrum . 42 5.4 Further reading . 44 6 Quantum stochastic methods 47 6.1 Quantum jump formalism . 47 iii iv CONTENTS 6.2 Heisenberg-Langevin equations . 49 6.3 Fluctuation dissipation theorem . 51 6.4 Further reading . 52 7 Cavity Quantum Electrodynamics 57 7.1 The Purcell effect in a 1D model cavity . 57 7.2 Weak to strong coupling via density matrices . 61 7.3 Examples of Cavity QED systems . 63 7.4 Further reading . 67 8 Superradiance 69 8.1 Simple density matrix equation for collective emission . 69 8.2 Beyond the simple model . 74 8.3 Further reading . 78 9 The Dicke model 79 9.1 Phase transitions, spontaneous superradiance . 79 9.2 No-go theorem: no vacuum instability . 81 9.3 Radiation in a box; restoring the phase transition . 82 9.4 Dynamic superradiance . 83 9.5 Further Reading . 86 10 Lasers and micromasers 89 10.1 Density matrix equations for a micromaser and a laser . 89 10.2 Laser rate equations . 92 10.3 Laser Linewidth . 93 10.4 Further reading . 96 11 More on lasers 99 11.1 Density matrix equation . 99 11.2 Spontaneous emission, noise, and β parameter . 106 11.3 Single atom lasers . 109 11.4 Further reading . 112 12 Three levels, and coherent control 113 12.1 Semiclassical introduction . 113 12.2 Coherent evolution alone; why does EIT occur . 117 12.3 Dark state polaritons . 117 12.4 Further reading . 120 Bibliography 123 Introduction The title quantum optics covers a large range of possible courses, and so this introduction intends to explain what this course does and does not aim to provide. Regarding the negatives, there are several things this course deliberately avoids: • It is not a course on quantum information theory. Some basic notions of coherent states and entanglement will be assumed, but will not be the focus. • It is not a course on relativistic gauge field theories; the majority of solid state physics does not require covariant descriptions, and so it is generally not worth paying the price in complexity of using a manifestly covariant formulation. • As far as possible, it is not a course on semiclassical electromagnetism. While at times radiation will be treated classically, this will generally be for comparison to a full quantum treatment, or where such an approximation is valid (for at least part of the radiation). Regarding the positive aims of this course, they are: to discuss how to model the quantum behaviour of coupled light and matter; to introduce some simple models that can be used to describe such systems; to dis- cuss methods for open quantum systems that arise naturally in the context of coupled light and matter; and to discuss some of the more interesting phenomena which may arise for matter coupled to light. Semiclassical be- haviour will be discussed in some sections, both because an understanding of semiclassical behaviour (i.e. classical radiation coupled to quantum me- chanical matter) is useful to motivate what phenomena might be expected; and also as comparison to the semiclassical case is important to see what new physics arises from quantised radiation. The kind of quantum optical systems discussed will generally consist of one or many few-level atoms coupled to one quantised radiation fields. Realisations of such systems need not involve excitations of real atoms, but can instead be artificial atoms, i.e. well defined quantum systems with dis- crete level spectra which couple to the electromagnetic field. Such concepts therefore apply to a wide variety of systems, and a variety of character- istic energies of electromagnetic radiation. Systems currently studied ex- perimentally include: real atomic transitions coupled to optical cavities[1]; v vi INTRODUCTION Josephson junctions in microwave cavities (waveguides terminated by re- flecting boundaries)[2, 3]; Rydberg atoms (atoms with very high princi- ple quantum numbers, hence small differences of energy levels) in GHz cavities[4]; and solid state excitations, 1i.e. excitons or trions localised in quantum dots, coupled to a variety of optical frequency cavities, includ- ing simple dielectric contrast cavities, photonic band gap materials, and whispering gallery modes in disks[5]. These different systems provide different opportunities for control and measurement; in some cases one can probe the atomic state, in some cases the radiation state. To describe experimental behaviour, one is in gen- eral interested in calculating a response function, relating the expected outcome to the applied input. However, to understand the predicted be- haviour, it is often clearer to consider the evolution of quantum mechanical state; thus, both response functions and wavefunctions will be discussed. As such, the lectures will switch between Heisenberg and Schr¨odingerpic- tures frequently according to which is most appropriate. When considering open quantum systems, a variety of different approaches; density matrix equations, Heisenberg-Langevin equations and their semiclassical approxi- mations, again corresponding to both Schr¨odinger and Heisenberg pictures. The main part of this course will start with the simplest case of a single two-level atom, and discuss this in the context of one or many quantised radiation modes. The techniques developed in this will then be applied to the problem of many two-level atoms, leading to collective effects. The techniques of open quantum systems will also be applied to describing las- ing, focussing on the \more quantum" examples of micromasers and single atom lasers. The end of the course will consider atoms beyond the two-level approximation, illustrating what new physics may arise. Separate to this main discussion, the first two lectures stand alone in discusing where the simple models of coupled light and matter used in the rest of the course come from, in terms of the quantised theory of electromagnetism. Lecture 1 Quantisation of electromagnetism in the Coulomb gauge Our aim is to write a theory of quantised radiation interacting with quan- tised matter fields. Such a theory, e.g. the Jaynes-Cummings model (see next lecture) has an intuitive form: X y X z + HJ:C: = !kakak + iσi + gi;kσi ak + H.c. : (1.1) k i;k y The operator ak creates \a photon" in the mode with wavevector k, and so this Hamiltonian describes a process where a two-level system can change its state with the associated emission or absorption of a photon. The term y !kakak then gives the total energy associated with occupation of the mode with energy !k. While the rest of the course is dedicated to studying such models of coupled light-matter system, this (and in part the next) lecture will show the relation between such models and the classical electromag- netism of Maxwell's equations. To reach this destination, we will follow the route of canonical quan- tisation; our first aim is therefore to write a Lagrangian in terms of only relevant variables. Relevant variables are those where both the variable and its time derivative appear in the Lagrangian; if the time derivative does not appear, then we cannot define the canonically conjugate momentum, and so cannot enforce canonical commutation relations. The simplest way of writing the Lagrangian for electromagnetism contains irrelevant variables | i.e. the electric scalar potential φ and gauge of the vector potential A; that irrelevant variables exist is due to the gauge invariance of the theory. Since we are not worried about preserving manifest Lorentz covariance, we are free to solve this problem in the simplest way | eliminating the unnecessary variables. 1 2 LECTURE 1. QUANTISATION OF ELECTROMAGNETISM 1.1 Revision: Lagrangian for electromagnetism To describe matter interacting with radiation, we wish to write a La- grangian whose equations of motion will reproduce Maxwell's and Lorentz's equations: r × B = µ0J + µ0"0E_ r · E = ρ/ε0 (1.2) r · B = 0 r × E = −B_ (1.3) mα¨rα = qα[E(rα) + r_ α × B(rα)]: (1.4) Equations (1.3) determine the structure of the fields, not their dynamics, and are immediately satisfied by defining B = r×A and E = −∇φ−A_ . Let us suggest the form of Lagrangian L that leads to Eq. (1.2) and Eq. (1.4): Z X 1 "0 X L = m r_ 2 + dV E2 − c2B2 + q [r_ · A(r ) − φ(r )] : 2 α α 2 α α α α α α (1.5) Here, the fields E and B should be regarded as functionals of φ and A. Note also that in order to be able to extract the Lorentz force acting on individual charges, the currents and charge
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